联合插值限制的实值函数的L_2逼近

L_2- Approximation of Real-Valued Functions with Simultaneous Interpolatory Constraints

我们构造一个ｍ次多项式ｐｍ，ｎ，它是一个在给定的几个不同的结点上对已给实函数ｆ∈Ｌ＾２１，∞进行联合插值，满足Ｐｍ，ｎ（ｘｉ）＝ｆ（ｘｉ），Ｐｍ，ｎ’（Ｘｉ），ｉ，…，ｎ，在Ｌ２范数下，在ｆ的所有同样性质的插值多项式中，它又是ｆ的最佳逼近，并且得到当ｆ∈Ｃ「ａ，ｂ」，ｍ→∞，‖Ｐｍ，ｎ－ｆ‖→０。

We construct the polynomial pm,n of degree m which simultanious interpolates a given real-valued function f∈L1,∞2[a,b] at pre-assigned n distinct nodes satisfies: p(x:) = f(xi). p1(xi) = f1(xi)i=1,…, and is the best approximant to fin the L2-sense over all polynomials of degree≤m with the same interpolatory character. It is shown that the L2- error ||pm,n -f||→ 0asm→ ∞if f∈C[a,b].